We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math,
2016). If
M is a compact metric space,
\(c : M\times M \rightarrow \mathbb {R}\) a continuous cost function and
\(\lambda \in (0,1)\), the unique solution to the discrete
\(\lambda \)-discounted equation is the only function
\(u_\lambda : M\rightarrow \mathbb {R}\) such that
$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$
We prove that there exists a unique constant
\(\alpha \in \mathbb {R}\) such that the family of
\(u_\lambda +\alpha /(1-\lambda )\) is bounded as
\(\lambda \rightarrow 1\) and that for this
\(\alpha \), the family uniformly converges to a function
\(u_0 : M\rightarrow \mathbb {R}\) which then verifies
$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$
The proofs make use of Discrete Weak KAM theory. We also characterize
\(u_0\) in terms of Peierls barrier and projected Mather measures.